Yahoo Answers: Answers and Comments for Show that any group of order 35 is cyclic.? [Mathematics]
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From Kamal S
enUS
Tue, 04 Jun 2013 08:53:13 +0000
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Yahoo Answers: Answers and Comments for Show that any group of order 35 is cyclic.? [Mathematics]
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From phyllys: right this is an oldie yet goodie: what's ...
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https://answers.yahoo.com/question/index?qid=20130604085313AAWhGAn
Mon, 26 Dec 2016 17:34:09 +0000
right this is an oldie yet goodie: what's pink and commutes? An Abelian Grape! Get it? An Abelian Grape, Abelian group? Commutative property? ok, perhaps it wasn't that sturdy...

From kb: Let G be a group of order 35 = 5 * 7.
So, G...
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Tue, 04 Jun 2013 09:33:30 +0000
Let G be a group of order 35 = 5 * 7.
So, G has pSylow subgroups of orders p = 5, 7; let m and n (respectively) be the number of these subgroups.
Then, we have
(i) m5 and m = 1 (mod 7) ==> m = 1.
(ii) n7 and n = 1 (mod 5) ==> n = 1.
Hence, both subgroups are normal in G.
Being of prime order, these are cyclic of orders 5 and 7, which has trivial intersection.
Therefore, G ≅ Z5 x Z7 ≅ Z35, which is cyclic.
I hope this helps!